To understand how the function \(f(x) = \sqrt[3]{x}\) is transformed into \(g(x) = 2 f(x-3)\), let's break it down step-by-step.
1. Parent Function: The parent function here is \(f(x) = \sqrt[3]{x}\). This function represents the cube root of \(x\).
2. Horizontal Shift: The transformation \(f(x-3)\) means we are shifting the function \(f(x)\) horizontally. Specifically, \(f(x-3)\) shifts the graph of \(f(x)\) 3 units to the right.
3. Vertical Stretch: The transformation \(2 f(x)\) implies that we are stretching the function vertically by a factor of 2. This means that the y-values of the function are doubled.
Now, let’s detail the transformation step-by-step:
### Step-by-Step Transformation:
#### Step 1: Horizontal Shift
Take the function \(f(x) = \sqrt[3]{x}\) and transform it to \(f(x-3)\):
[tex]\[ f(x-3) = \sqrt[3]{x-3} \][/tex]
This shifts the graph of \(f(x)\) 3 units to the right.
#### Step 2: Vertical Stretch
Next, we apply the vertical stretch to the result:
[tex]\[ g(x) = 2f(x-3) = 2 \sqrt[3]{x-3} \][/tex]
This stretches the graph of \( \sqrt[3]{x-3} \) vertically by a factor of 2.
### Graph of \(g(x)\)
- Horizontal Shift: The original cube root function \(\sqrt[3]{x}\) is centered at the origin. The transformation \(\sqrt[3]{x-3}\) moves the center to (3,0).
- Vertical Stretch: The points on the graph now have their y-coordinates doubled.
Here are a few points for clarity:
- The point (0, 0) on \( \sqrt[3]{x} \) becomes (3, 0) on \( \sqrt[3]{x-3} \).
- The point (1, 1) on \( \sqrt[3]{x} \) becomes (4, 2) on \( 2\sqrt[3]{x-3} \).
- The point (-1, -1) on \( \sqrt[3]{x} \) becomes (2, -2) on \( 2\sqrt[3]{x-3} \).
Therefore, the graph of \(g(x) = 2 \sqrt[3]{x-3} \) will be a horizontally shifted (to the right by 3 units) and vertically stretched version (by a factor of 2) of the parent function \(f(x) = \sqrt[3]{x}\).
To choose the correct graph, look for a graph that is:
- Shifted to the right by 3 units.
- Stretched vertically by a factor of 2.
This should help you identify the correct graph.
